a ladder 10 ft long rests against a vertical wall. if the bottom of the ladder slides away from the wall at a speed of 2 ft/s how fast is the angle between the top of the ladder and the wall changing when the angle is pi/4 radians?
a ladder 10 ft long rests against a vertical wall. if the bottom of the ladder slides away from the wall at a speed of 2 ft/s how fast is the angle between the top of the ladder and the wall changing when the angle is pi/4 radians?
I might be wrong but I think this is right.
If you think of the angle $\displaystyle \theta$ the only thing way of measuring it is by the hypotenuse. So cosine is useful, but how do you find the adjacent side?
well the hypotenuse = 10
the adjacent side is some value let's say C plus it's growing by 2ft/sec. So mathematically that means:
$\displaystyle 2\frac{\text{ft}}{\text{sec}}*\text{time}+ C$
aka: $\displaystyle 2t+C$
so then: $\displaystyle cos\theta=\frac{2t+C}{10}$
So now you want to see how much the angle theta is changing as time goes by, so you take the derivative in terms of t.
$\displaystyle \frac{d}{dt}(cos\theta)=\frac{d}{dt}(\frac{2t+C}{1 0})$
Can you take it from there?